Properties

Degree 2
Conductor $ 2^{5} \cdot 3^{3} \cdot 7 $
Sign $-0.707 - 0.707i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.317i·5-s + i·7-s − 1.41·11-s − 4.44·13-s + 5.97i·17-s + 2.44i·19-s + 5.65·23-s + 4.89·25-s − 5.51i·29-s − 6.44i·31-s − 0.317·35-s + 9·37-s − 7.24i·41-s + 6.34i·43-s + 10.8·47-s + ⋯
L(s)  = 1  + 0.142i·5-s + 0.377i·7-s − 0.426·11-s − 1.23·13-s + 1.44i·17-s + 0.561i·19-s + 1.17·23-s + 0.979·25-s − 1.02i·29-s − 1.15i·31-s − 0.0537·35-s + 1.47·37-s − 1.13i·41-s + 0.968i·43-s + 1.58·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $-0.707 - 0.707i$
motivic weight  =  \(1\)
character  :  $\chi_{6048} (2591, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 6048,\ (\ :1/2),\ -0.707 - 0.707i)$
$L(1)$  $\approx$  $1.015344525$
$L(\frac12)$  $\approx$  $1.015344525$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 - iT \)
good5 \( 1 - 0.317iT - 5T^{2} \)
11 \( 1 + 1.41T + 11T^{2} \)
13 \( 1 + 4.44T + 13T^{2} \)
17 \( 1 - 5.97iT - 17T^{2} \)
19 \( 1 - 2.44iT - 19T^{2} \)
23 \( 1 - 5.65T + 23T^{2} \)
29 \( 1 + 5.51iT - 29T^{2} \)
31 \( 1 + 6.44iT - 31T^{2} \)
37 \( 1 - 9T + 37T^{2} \)
41 \( 1 + 7.24iT - 41T^{2} \)
43 \( 1 - 6.34iT - 43T^{2} \)
47 \( 1 - 10.8T + 47T^{2} \)
53 \( 1 - 11.9iT - 53T^{2} \)
59 \( 1 + 5.83T + 59T^{2} \)
61 \( 1 + 10.4T + 61T^{2} \)
67 \( 1 - 5.79iT - 67T^{2} \)
71 \( 1 + 5.65T + 71T^{2} \)
73 \( 1 + 9.79T + 73T^{2} \)
79 \( 1 - 14.3iT - 79T^{2} \)
83 \( 1 + 8.02T + 83T^{2} \)
89 \( 1 + 15.4iT - 89T^{2} \)
97 \( 1 + 14.2T + 97T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.230054114844299567113760346207, −7.64539829460196804976377160114, −7.04695972018720291151006690361, −5.99165718985852923205578687662, −5.71007051929001492565790794333, −4.60778296023770561200094034392, −4.11331295231782530461558292651, −2.86338300770114560825383493215, −2.41956412658744092732424882135, −1.19241163964774588089922152663, 0.27461725918572087729231263113, 1.33541270929710765694681835280, 2.80200742545663678989121902775, 2.93986100037202406913883720451, 4.38517056326215640706808810316, 4.95389391468189891353637781172, 5.38689188521068343833080086417, 6.66964989575803218073464380121, 7.11773521575869507755976082858, 7.61962157045837616922071999032

Graph of the $Z$-function along the critical line